[zxv]

Result

An explicit six-source depth-two block attains the unsigned dirty connector’s complete-call floor in every ring characteristic $c\geq8$. Concatenation gives

$$ N(c)\geq2c+2\lfloor c/2\rfloor. $$

That is $3c$ sources for even $c$ and $3c-1$ for odd $c$, compared with the lower bound $\lfloor5c/2\rfloor$ in Nick Petalas’s PR #6035. For example, the attaining source counts rise from 20 to 24 at characteristic eight, 22 to 26 at nine, and 25 to 30 at ten. This answers that predecessor’s first open question about a six-source depth-two block.

The 54-event template uses each source nine times. Every one of the 63 nonzero binary masks is an exact integer sum of two lifted prefixes. Replacing the first source event past each cut with a run of $c-8$ copies transfers every identity to characteristic $c$. The full word preserves all immutable sources and arbitrary dirty outputs, restores its arbitrary dirty connector, calls every source and output operator exactly $c$ times, and attains

$$ c\bigl(k+2^k-1\bigr) $$

complete calls. The proof rederives the required lift, concatenation and deletion implications. Its integer-lift method and block framework remain credited to PR #6035 and PR #4248.

Significance for researchers

Six sources per two prefix reads improve the uniform characteristic-dependent construction while retaining the complete dirty interface. This is an affirmative finite construction with an all-characteristic lifting proof. It changes no characteristic-two BIT word, native supplier, selected frontier or κ. Input admission, physical call implementations, coefficient preparation, dispatch, hosting and the consumer’s complete bill remain separate.

Validation

CPython 3.12.8 normal and optimized registered replays pass all 63 integer identities, every lift boundary and mask for characteristics 8 through 40, 35,478 literal calls in complete six- and twelve-source symbolic words, noncommuting two-by-two matrix controls over prime/composite characteristic rings, and 1,385 composed masks. Missing calls, changed labels/readouts and exceeded depth/run budgets reject. The maximal exponentially large output family is proved by block decomposition, rather than materialized.

Committed-byte preflight and the official nine-file package export/integrity check pass. Search is not trusted by replay. The optional full-history check is unavailable in this shallow checkout; all pins are at the reachable base. The navigation-only repair from PR #6050 is included; no frozen package or selected record is modified.

Open next questions

Can six sources fit depth two in characteristic five, six or seven? Can seven sources fit depth two, or an odd-depth block remove the one-source loss for odd characteristics? First produce an exact template or scoped obstruction. Bounded failed discovery searches exclude nothing.

Does one actual consumer benefit after all physical operations and restoration are paid? Pin its source/consumer contract and full invoice before extending the construction.

Prepared by zxv with substantial OpenAI Codex assistance. Predecessor attribution and exact sources are retained; no independent human or proof-assistant review is claimed.

Final export validation at f92bd41f60564175bd68f944e9d6450da8965734: official committed export, integrity and declared-input coverage pass. A fresh optimized Git-free replay with empty PATH and read-only exported files passes; every exported byte remains unchanged.

Hosted shared planning and the scientific connector-six-source-lift replay both pass at f92bd41f60564175bd68f944e9d6450da8965734 in run 40628906429. Unrelated archived source-pin jobs remain separate from this scientific result.